${Bounded composition operators on the spaces $H_{\omega,p

Let $\mathbb{D}$ denote the unit disk in the complex plane $\mathbb{C}$, and $\mathcal{H}(\mathbb{D})$ the class of all analytic functions on $\mathbb{D}$. Take a positive function $\omega\in C^2[0,1)$, and call such $\omega$ a weight function. For $p 0$, a function $f\in\mathcal{H}(\mathbb{D})$ is said to belong to the space $H_{\omega,p}$, if $$\|f\|_{\omega,p}^p=|f(0)|^p+p^2\int_\mathbb{D}|f(z)|^{p-2}|f (z)|^2\omega(|z|)dm(z) \infty,$$ where $dm$ stands for the normalized Lebesgue area measure on $\mathbb{D}$. We study the boundedness of the composition operators on the space $H_{\omega,p}$ when $\omega$ is an $(ii)$-admissible weight. In particular, we use a family of functions and generalized Nevanlinna counting functions for our characterization.